- the very special and untouched case of Cuba in terms of the Zika epidemics, thanks to a long term policy fighting mosquitoes at all levels of the society;
- an impressive map of the human cortex, which statistical analysis would be fascinating;
- an excerpt from Nature 13 August 1966 where the Poisson distribution was said to describe the distribution of scores during the 1966 World Cup;
- an analysis of a genetic experiment on evolution involving 50,000 generations (!) of Escherichia coli;
- a look back at the great novel Flowers for Algernon, novel I read eons ago;
- a Nature paper on the first soft robot, or octobot, along with some easier introduction, which did not tell which kind of operations could be accomplished by such a robot;
- a vignette on a Science paper about the interaction between honey hunters and hunting birds, which I also heard depicted on the French National Radio, with an experiment comparing the actual hunting (human) song, a basic sentence in the local language, and the imitation of the song of another bird. I could not understand why the experiment did not include hunting songs from other hunting groups, as they are highly different but just as effective. It would have helped in understanding how innate the reaction of the bird is;
- another literary entry at the science behind Mary Shelley’s Frankenstein;
- a study of the Mathematical Genealogy Project in terms of the few mathematicians who started most genealogies of mathematicians, including d’Alembert, advisor to Laplace of whom I am one of the many descendants, although the finding is not that astounding when considering usual genealogies where most branches die off and the highly hierarchical structure of power in universities of old.
Archive for Pierre Simon de Laplace
An interesting question from a reader of the Bayesian Choice came out on X validated last week. It was about Laplace’s succession rule, which I found somewhat over-used, but it was nonetheless interesting because the question was about the discrepancy of the “non-informative” answers derived from two models applied to the data: an Hypergeometric distribution in the Bayesian Choice and a Binomial on Wikipedia. The originator of the question had trouble with the difference between those two “non-informative” answers as she or he believed that there was a single non-informative principle that should lead to a unique answer. This does not hold, even when following a reference prior principle like Jeffreys’ invariant rule or Jaynes’ maximum entropy tenets. For instance, the Jeffreys priors associated with a Binomial and a Negative Binomial distributions differ. And even less when considering that there is no unity in reaching those reference priors. (Not even mentioning the issue of the reference dominating measure for the definition of the entropy.) This led to an informative debate, which is the point of X validated.
On a completely unrelated topic, the survey ship looking for the black boxes of the crashed EgyptAir plane is called the Laplace.
Anindya Bhadra, Jyotishka Datta, Nick Polson and Brandon Willard have arXived this morning a short paper on global-local mixtures. Although the definition given in the paper (p.1) is rather unclear, those mixtures are distributions of a sample that are marginals over component-wise (local) and common (global) parameters. The observations of the sample are (marginally) exchangeable if not independent.
“The Cauchy-Schlömilch transformation not only guarantees an ‘astonishingly simple’ normalizing constant for f(·), it also establishes the wide class of unimodal densities as global-local scale mixtures.”
The paper relies on the Cauchy-Schlömilch identity
a self-inverse function. This generic result proves helpful in deriving demarginalisations of a Gaussian distribution for densities outside the exponential family like Laplace’s. (This is getting very local for me as Cauchy‘s house is up the hill, while Laplace lived two train stations away. Before train was invented, of course.) And for logistic regression. The paper also briefly mentions Etienne Halphen for his introduction of generalised inverse Gaussian distributions, Halphen who was one of the rare French Bayesians, worked for the State Electricity Company (EDF) and briefly with Lucien Le Cam (before the latter left for the USA). Halphen introduced some families of distributions during the early 1940’s, including the generalised inverse Gaussian family, which were first presented by his friend Daniel Dugué to the Académie des Sciences maybe because of the Vichy racial laws… A second result of interest in the paper is that, given a density g and a transform s on positive real numbers that is decreasing and self-inverse, the function f(x)=2g(x-s(x)) is again a density, which can again be represented as a global-local mixture. [I wonder if these representations could be useful in studying the Cauchy conjecture solved last year by Natesh and Xiao-Li.]
Following my earlier post [induced by browsing X validated], on the strange property that the product of a Normal variate by an Exponential variate is a Laplace variate, I got contacted by Peng Ding from UC Berkeley, who showed me how to derive the result by a mere algebraic transform, related with the decomposition
(X+Y)(X-Y)=X²-Y² ~ 2XY
when X,Y are iid Normal N(0,1). Peng Ding and Joseph Blitzstein have now arXived a note detailing this derivation, along with another derivation using the moment generating function. As a coincidence, I also came across another interesting representation on X validated, namely that, when X and Y are Normal N(0,1) variates with correlation ρ,
XY ~ R(cos(πU)+ρ)
with R Exponential and U Uniform (0,1). As shown by the OP of that question, it is a direct consequence of the decomposition of (X+Y)(X-Y) and of the polar or Box-Muller representation. This does not lead to a standard distribution of course, but remains a nice representation of the product of two Normals.
When browsing X validated the other day [translate by procrastinating!], I came upon the strange property that the marginal distribution of a zero mean normal variate with exponential variance is a Laplace distribution. I first thought there was a mistake since we usually take an inverse Gamma on the variance parameter, not a Gamma. But then the marginal is a t distribution. The result is curious and can be expressed in a variety of ways:
The OP was asking for a direct proof of the result and I eventually sorted it out by a series of changes of variables, although there exists a much more elegant and general proof by Mike West, then at the University of Warwick, based on characteristic functions (or Fourier transforms). It reminded me that continuous, unimodal [at zero] and symmetric densities were necessary scale mixtures [a wee misnomer] of Gaussians. Mike proves in this paper that exponential power densities [including both the Normal and the Laplace cases] correspond to the variances having an inverse positive stable distribution with half the power. And this is a straightforward consequence of the exponential power density being proportional to the Fourier transform of a stable distribution and of a Fubini inversion. (Incidentally, the processing times of Biometrika were not that impressive at the time, with a 2-page paper submitted in Dec. 1984 published in Sept. 1987!)
This is a very nice and general derivation, but I still miss the intuition as to why it happens that way. But then, I know nothing, and even less about products of random variates!
Looking at the Family Tree application (I discovered via Peter Coles’ blog), I just found out that I was Laplace’s [academic] great-great-great-great-great-great-great-grand-child! Through Poisson and Chasles. Going even further, as Simeon Poisson was also advised by Lagrange, my academic lineage reaches Euler and the Bernoullis. Pushing always further, I even found William of Ockham along one of the “direct” branches! Amazing ancestry, to which my own deeds pay little homage if any… (However, I somewhat doubt the strength of the links for the older names, since pursuing them ends up at John the Baptist!)
I wonder how many other academic descendants of Laplace are alive today. Too bad Family Tree does not seem to offer this option! Given the longevity of both Laplace and Poisson, they presumably taught many students, which means a lot of my colleagues and even of my Bayesian colleagues should share the same illustrious ancestry. For instance, I share part of this ancestry with Gérard Letac. And both Jean-Michel Marin and Arnaud Guillin. Actually, checking with the Mathematics Genealogy Project, I see that Laplace had… one student!, but still a grand total of [at least] 85,738 descendants… Incidentally, looking at the direct line, most of those had very few [recorded] descendants.